3.5.30 \(\int \frac {\sqrt {d+e x}}{\sqrt {-2 x-3 x^2}} \, dx\) [430]

Optimal. Leaf size=53 \[ -\frac {2 \sqrt {d+e x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {-x}\right )|\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {1+\frac {e x}{d}}} \]

[Out]

-2/3*EllipticE(1/2*6^(1/2)*(-x)^(1/2),1/3*6^(1/2)*(e/d)^(1/2))*(e*x+d)^(1/2)*3^(1/2)/(1+e*x/d)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {728, 12, 113, 111} \begin {gather*} -\frac {2 \sqrt {d+e x} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{2}} \sqrt {-x}\right )|\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {\frac {e x}{d}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/Sqrt[-2*x - 3*x^2],x]

[Out]

(-2*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[3/2]*Sqrt[-x]], (2*e)/(3*d)])/(Sqrt[3]*Sqrt[1 + (e*x)/d])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2*(Sqrt[e]/b)*Rt[-b/
d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[
d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !LtQ[-b/d, 0]

Rule 113

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x]*(Sqrt[
1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)])), Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /
; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 728

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1
+ (c/b)*x]), x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4] && LtQ[
c, 0] && RationalQ[b]

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x}}{\sqrt {-2 x-3 x^2}} \, dx &=\int \frac {\sqrt {d+e x}}{\sqrt {2} \sqrt {-x} \sqrt {1+\frac {3 x}{2}}} \, dx\\ &=\frac {\int \frac {\sqrt {d+e x}}{\sqrt {-x} \sqrt {1+\frac {3 x}{2}}} \, dx}{\sqrt {2}}\\ &=\frac {\sqrt {d+e x} \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {-x} \sqrt {1+\frac {3 x}{2}}} \, dx}{\sqrt {2} \sqrt {1+\frac {e x}{d}}}\\ &=-\frac {2 \sqrt {d+e x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {-x}\right )|\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {1+\frac {e x}{d}}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(117\) vs. \(2(53)=106\).
time = 1.78, size = 117, normalized size = 2.21 \begin {gather*} \frac {2 \sqrt {-\frac {d}{e}} (2+3 x) (d+e x)-2 d \sqrt {9+\frac {6}{x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {d}{e}}}{\sqrt {x}}\right )|\frac {2 e}{3 d}\right )}{3 \sqrt {-\frac {d}{e}} \sqrt {-x (2+3 x)} \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]/Sqrt[-2*x - 3*x^2],x]

[Out]

(2*Sqrt[-(d/e)]*(2 + 3*x)*(d + e*x) - 2*d*Sqrt[9 + 6/x]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[ArcSin[Sqrt[-(d/e)
]/Sqrt[x]], (2*e)/(3*d)])/(3*Sqrt[-(d/e)]*Sqrt[-(x*(2 + 3*x))]*Sqrt[d + e*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(214\) vs. \(2(45)=90\).
time = 0.45, size = 215, normalized size = 4.06

method	result	size
default	\(-\frac {2 \sqrt {e x +d}\, \sqrt {-x \left (2+3 x \right )}\, d \sqrt {\frac {e x +d}{d}}\, \sqrt {-\frac {\left (2+3 x \right ) e}{3 d -2 e}}\, \sqrt {-\frac {e x}{d}}\, \left (3 d \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d -2 e}}\right )-2 \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d -2 e}}\right ) e -3 \EllipticE \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d -2 e}}\right ) d +2 \EllipticE \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d -2 e}}\right ) e \right )}{3 e x \left (3 e \,x^{2}+3 d x +2 e x +2 d \right )}\)	\(215\)
elliptic	\(\frac {\sqrt {-x \left (2+3 x \right ) \left (e x +d \right )}\, \left (\frac {2 d^{2} \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x +\frac {2}{3}}{-\frac {d}{e}+\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {2}{3}\right )}}\right )}{e \sqrt {-3 e \,x^{3}-3 d \,x^{2}-2 e \,x^{2}-2 d x}}+\frac {2 d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x +\frac {2}{3}}{-\frac {d}{e}+\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}+\frac {2}{3}\right ) \EllipticE \left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {2}{3}\right )}}\right )-\frac {2 \EllipticF \left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {2}{3}\right )}}\right )}{3}\right )}{\sqrt {-3 e \,x^{3}-3 d \,x^{2}-2 e \,x^{2}-2 d x}}\right )}{\sqrt {-x \left (2+3 x \right )}\, \sqrt {e x +d}}\)	\(285\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)/(-3*x^2-2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*(e*x+d)^(1/2)*(-x*(2+3*x))^(1/2)*d*((e*x+d)/d)^(1/2)*(-(2+3*x)*e/(3*d-2*e))^(1/2)*(-e*x/d)^(1/2)*(3*d*Ell
ipticF(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d-2*e))^(1/2))-2*EllipticF(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d-2*e))^(1/2
))*e-3*EllipticE(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d-2*e))^(1/2))*d+2*EllipticE(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*
d-2*e))^(1/2))*e)/e/x/(3*e*x^2+3*d*x+2*e*x+2*d)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(-3*x^2-2*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x*e + d)/sqrt(-3*x^2 - 2*x), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(-3*x^2-2*x)^(1/2),x, algorithm="fricas")

[Out]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d + e x}}{\sqrt {- x \left (3 x + 2\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)/(-3*x**2-2*x)**(1/2),x)

[Out]

Integral(sqrt(d + e*x)/sqrt(-x*(3*x + 2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(-3*x^2-2*x)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(x*e + d)/sqrt(-3*x^2 - 2*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {d+e\,x}}{\sqrt {-3\,x^2-2\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)/(- 2*x - 3*x^2)^(1/2),x)

[Out]

int((d + e*x)^(1/2)/(- 2*x - 3*x^2)^(1/2), x)

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